A Simple Example (Exponential)

say X₁, .., X_n iid with f(x)=βexp(-βx), x>0 and we want to test H₀:β=β₀ vs H_a:β≠β₀

Wald type test

we can often find such a test by first finding the method of moments estimator:

so if 1/X̅ is close to β₀ we should accept the null hypothesis, otherwise we should reject it. This is of course equivalent toX̅ is close to 1/β₀. Now

so we reject H₀ if √n|X̅-1/β₀|/(1/β₀)=√n|β₀X̅-1|>qnorm(1-α/2) =z_α/2

LRT test

and we reject H₀ if -2logλ(x)>qchisq(1-α,1)

How about using the LRT without the chisquare approximation? As always we have

"reject H₀ if λ(x) is small"

iff

"reject H₀ if -2logλ(x) is large"

iff

"reject H₀ ifX̅ is small or large" (see exp.ex(3))

so we reject H₀ ifX̅<cv₁ orX̅>cv₂.

Now

and clearly cv₂=qgamma(1-α/2,n,β₀)/n

This solution only works because we know the distribution of a sum of exponential variables (gamma), in another example we might be stuck with using the chisquare approximation.

Which test is better?
for that we need to find the power of the tests:

and

The power curves are shown in exp.ex(5)